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साबित करें कि `2sin^(2) theta + 4 cos ( theta + alpha) sin alpha sin theta+ cos 2 ( alpha + theta) , theta ` से स्वतंत्र है |
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Answer» `2sin^(2) theta + 4 cos (theta + alpha) sin alpha sin theta +cos 2 ( alpha+ theta)` `= 2 sin^(2) theta + 2 cos ( theta + alpha) 2sinalphasin theta + cos 2 ( alpha + theta)` `= 2 sin^(2) theta+ 2 cos (theta + alpha) [cos ( theta - alpha) -cos ( theta + alpha) ] + cos 2 ( alpha+ theta)` `= 2 sin^(2) theta+ 2 cos (theta + alpha) cos ( theta - alpha) -2cos^(2) ( theta + alpha) + cos 2 ( alpha+ theta)` `=2 sin^(2) theta+2(cos^(2) theta-sin^(2) alpha) - 2 cos^(2) ( theta + alpha) + [2 cos^(2) ( alpha+ theta) -1]` `= 2 sin^(2) theta + 2 cos^(2) theta - 2sin^(2) alpha -1 ` `= 2( sin ^(2) theta + cos^(2) theta ) - 2 sin^(2) alpha-1 ` `= 2-2-sin^(2) alpha-1= 1- 2 sin^(2) alpha = cos2 alpha` जो कि `theta ` से स्वतंत्र है । |
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